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16
JanuszWalczak,AnnaRomanowska
t
y
(
j
ω
,
t
)
=
∫
h
(
t
−
ζ
,
ζ
)
e
−
j
ωζ
d
ζ
,
(9)
−
∞
introducingavalue1=e
j
ω
t·e-j
ω
tintoequation(9)resultsin:
t
y
(
j
ω
,
t
)
=
e
j
ω
t
∫
h
(
t
,
ζ
)
e
−
j
ω
(
t
−
ζ
)
d
ζ
.
(10)
−
∞
Asinthecaseofstationarysystems,frequencycharacteristicisdefinedasaratioof
outputsignalandinputsignal:
H
(
j
ω
,
t
)
=
y
(
e
j
ω
j
ω
t
,
t
)
=
∞
∫
0
h
(
t
,
ζ
)
e
−
j
ω
(
t
−
ζ
)
d
ζ
.
(11)
Takingintoaccountthatt-
ζ
=
τ
andh(
τ
,t)=h(
τ
,t-
τ
)onecanobtainatransfer
functioninnormalform:
∞
H
(
j
ω
,
t
)
=
F
{
h
(
τ
,
t
)
}
=
∫
h
(
t
,
ζ
)
e
−
j
ω
(
t
−
ζ
)
d
ζ
.
(12)
0
3.DeterminationoffrequencyresponseofLTVsecondordersection
Thedetailedanalysisintimedomainforthreementionedearliercaseshasbeen
carriedoutinworks:[10],[11],[12].Thisanalysisallowstoobtainthegeneralized
formula(13)whichisthesectionresponsetoanyexcitationx(t).Basedonformula
(13)onecangettheimpulseresponseofLTVsection.Generally,thesolutionto
differentialequationofLTVsectionwithzeroinitialconditionsisexpressedby
[13]:
t
y
(
t
)
=
∫
A
[
q
2
(
t
)
q
1
(
ζ
)
−
q
1
(
t
)
q
2
(
ζ
)
]
e
−
a
(
ζ
)
e
a
(
t
)
x
(
τ
)
d
τ
,
(13)
0
where:A-constantcoefficientforeverycase,whichisdefinedasadeterminantof
WronskiMatrix(matrixoffundamentalsolutions)forsystemsI,IIandIII[10],
[11],[12],q1(t),q2(t)-fundamentalsolutionsforcasesI,IIandIII[10],[11],
[12].
Thecoefficientsandfunctionsinequations(13)aredefinedas:
CaseI(
ω
0
2(t)-var,
σ
0-const)[10]:
a
(
t
)
=
−
σ
0
ω
0
t
,
(14)
q
1
(
t
)
=
J
v
⎛
⎜
⎜
⎝
−
γ
2
C
e
−t
γ
2
⎞
⎟
⎟
⎠
,
(15)
q
2
(
t
)
=
Y
v
⎛
⎜
⎜
⎝
−
γ
2
C
e
−t
γ
2
⎞
⎟
⎟
⎠
,
(16)
